The final step in the FSSA algorithm is reconstruction of the functional singular spectrum analysis (fssa) objects into additive functional time series (fts) objects that capture different pieces of information of the original fts object. We then look at fts plots to analyze the data further.

We reconstruct for our multivariate Jambi NDVI and EVI data example using the following.

recon <- freconstruct(U = U, group = list(c(1),c(2,3),c(4)))

We can use suitable fts plots to view each reconstruction separately and obtain information about the orignal fts object.

The trend component for the NDVI data is best visualized using a 3D surface plot. This shows that in general, the pixel values tend to be be around 0.6 regardless of day which indicates higher levels of vegetation.

plot(recon[[1]],type = '3Dsurface',var=1,ylab = " ", xlab = "Pixel Value")

The season component for the NDVI data is also best visualized using a 3D surface plot. This shows the 6 month hot to cold and back to hot season oscillation that we are expecting.

plot(recon[[2]],type = '3Dsurface',var=1,ylab = " ", xlab = "Pixel Value")

The fourth component is best visualized with a 3D surface as well. This is the most interesting component as it uncovers a periodicity that was not obvious from before the analysis. We see that the density of the NDVI data undergoes a roughly decade long change from having a higher probability of observing higher pixel values to a lower probability of obseriving higher pixel values. This in turn means we went from a lower probability of observing lower pixel values to a higher probability of observing lower pixel values. This gives us a clue that in the past decade, there has been a loss in vegetation as captured by the NDVI data the cause of which needs to be investigated.

plot(recon[[3]],type = '3Dsurface',var=1,ylab = " ", xlab = "Pixel Value")

We see the same story manifest for the EVI data with the only difference being in the trend where we are still in general observing slightly higher pixel values with respect to this different index where the average value of the NDVI data set is higher (0.47) than the average value of the EVI data set (0.34).

plot(recon[[1]],type = '3Dsurface',var=2,ylab = " ", xlab = "Pixel Value")
plot(recon[[2]],type = '3Dsurface',var=2,ylab = " ", xlab = "Pixel Value")
plot(recon[[3]],type = '3Dsurface',var=2,ylab = " ", xlab = "Pixel Value")

This shows that the FSSA algorithm can not only offer information about fts objects that we expect to find, but it also can uncover interesting information that is not so obvious from fts plots of the original object or from common knowledge. Clearly there has been a loss of vegetation in the Jambi province in the last decade. The user can use this information to further explore the issue and try to figure out the source of such a loss of vegetation.